Step 3 : Find roots of the above equation, these will be the values of 'c'

\(f(x) = x^2 -2x + 4\) in [1, 5]

To find c, solve

\( f ' (x) = \dfrac {f(5) - f(1)}{5-1}\)

\(\Rightarrow 2x-2=\dfrac {(25-10+4)-(1-2+4)}{4}\)

\(\Rightarrow 2x-2=\dfrac {16}{4}\)

\(\Rightarrow 2x-2=4\)

\(\Rightarrow x=3\)

Since, \(3\in (1, 5)\)

\(\Rightarrow c = 3\)

Find the number 'c' that satisfies the conclusion of LMVT for following function:
\(f(x) = x^2 -2x + 4\) in [1, 5]

A

c = –1

.

B

c = 3

C

c = 4/7

D

c = 5/2

Option B is Correct

Newton - Raphson Method

It is a method used to approximate the roots of an equation. There are some equations whose exact roots are very hard to find, we apply this method to find an approximate root.

Consider the graph of a function \('f'\).

\(f(x) = 0\) has roots \(\alpha\) which we don't know.

Take any value of \(x\) say \(x=x_1\), draw tangent to the graph at \((x, f(x))\), let it cut the \(x\) axis at \(x=x_2\). Now again drawn a tangent at the point \((x_2, f(x_2))\). Let this intersect \(x\) axis at \(x_3\), observe that these values are approaching \(\alpha\) as we do these steps again and again,